Abstract
With respect to probabilistic mixtures of the strategies in non-cooperative games, quantum game theory provides guarantee of fixed-point stability, the so-called Nash equilibrium. This permits players to choose mixed quantum strategies that prepare mixed quantum states optimally under constraints. We show here that fixed-point stability of Nash equilibrium can also be guaranteed for pure quantum strategies via an application of the Nash embedding theorem, permitting players to prepare pure quantum states optimally under constraints.
This manuscript has been authored by UT-Battelle, LLC under Contract No. DE-AC05-00OR22725 with the U.S. Department of Energy. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes. The Department of Energy will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (http://energy.gov/downloads/doe-public-access-plan).
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Albers, S., Eilts, S., Even-Dar, E., Mansour, Y., Roditty, L.: On Nash equilibria for a network creation game. In: Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithm, pp. 89–98 (2006)
Fabrikant, A., Luthra, A., Menva, E., Papdimitriou, C., Shenker, S.: On a network creation game. In: Proceedings of the Twenty-Second Annual Symposium on Principles of Distributed Computing, pp. 347–351 (2003)
Liu, B., Dai, H., Zhang, M.: Playing distributed two-party quantum games on quantum networks. Quant. Inf. Process. 16, 290 (2017). https://doi.org/10.1007/s11128-017-1738-0
Scarpa, G.: Network games with quantum strategies. In: Sergienko, A., Pascazio, S., Villoresi, P. (eds.) QuantumComm 2009. LNICST, vol. 36, pp. 74–81. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-11731-2_10
Dasari, V., Sadlier, R.J., Prout, R., Williams, B.P., Humble, T.S.: Programmable multi-node quantum network design and simulation. In: SPIE Commercial+ Scientific Sensing and Imaging, pp. 98730B–98730B (2016)
Binmore, K.: Playing for Real. Oxford University Press, Oxford (2017)
Blaquiere, A.: Wave mechanics as a two-player game. In: Blaquiére, A., Fer, F., Marzollo, A. (eds.) Dynamical Systems and Microphysics, vol. 261, pp. 33–69. Springer, Vienna (1980). https://doi.org/10.1007/978-3-7091-4330-8_2
Meyer, D.: Quantum strategies. Phys. Rev. Lett. 82, 1052–1055 (1999)
Eisert, J., Wilkens, M., Lewenstien, M.: Quantum games and quantum strategies. Phys. Rev. Lett. 83, 3077 (1999)
Nash, J.: Equilibrium points in N-player games. Proc. Natl. Acad. Sci. USA 36, 48–49 (1950)
Kakutani, S.: A generalization of Brouwer’s fixed point theorem. Duke Math. J. 8(3), 457–459 (1941)
Glicksberg, I.L.: A further generalization of the Kakutani fixed point theorem, with application to Nash equilibrium points. Proc. Am. Math. Soc. 3, 170–174 (1952)
Nash, J.: The imbedding problem for Riemannian manifolds. Ann. Math. 63(1), 20–63 (1956)
Bengtsson, I., Zyczkowski, K.: Geometry of Quantum States: An Introduction to Quantum Entanglement, 1st edn. Cambridge University Press, Cambridge (2007)
Browuer, L.: Ueber eineindeutige, stetige transformationen von Flächen in sich. Math. Ann. 69 (1910)
Caratheodory, C.: Math. Ann. 64, 95 (1907). https://doi.org/10.1007/BF01449883
Kannai, Y.: An elementary proof of the no-retraction theorem. Am. Math. Mon. 88(4), 264–268 (1981)
Teklu, B., Olivares, S., Paris, M.: Bayesian estimation of one-parameter qubit gates. J. Phys. B: Atom. Mol. Opt. Phys. 42, 035502 (6pp) (2009)
Teklu, B., Genoni, M., Olivares, S., Paris, M.: Phase estimation in the presence of phase diffusion: the qubit case. Phys. Scr. T140, 014062 (3pp) (2010)
Khan, F.S., Solmeyer, N., Balu, R., Humble, T.S.: Quantum games: a review of the history, current state, and interpretation. Quant. Inf. Process. 17(11), 42 pp. Article ID 309. arXiv:1803.07919 [quant-ph]
Khan, F.S., Phoenix, S.J.D.: Gaming the quantum. Quant. Inf. Comput. 13(3–4), 231–244 (2013)
Khan, F.S., Phoenix, S.J.D.: Mini-maximizing two qubit quantum computations. Quant. Inf. Process. 12(12), 3807–3819 (2013)
Grover, L.K.: A fast quantum mechanical algorithm for database search. In: Proceedings of the 28th Annual ACM Symposium on the Theory of Computing, p. 212, May 1996
Williams, B.P., Britt, K.A., Humble, T.S.: Tamper-indicating quantum seal. Phys. Rev. Appl. 5, 014001 (2016)
Chitambar, E., Leung, D., Mančinska, L., Ozols, M., Winter, A.: Everything you always wanted to know about LOCC (but were afraid to ask). Commun. Math. Phys. 328(1), 303–326 (2014)
Farhi, E., Goldstone, J., Gutmann, S., Sipser, M.: Quantum computation by adiabatic evolution (preprint). https://arxiv.org/abs/quant-ph/0001106
McGeoch, C.C.: Adiabatic Quantum Computation and Quantum Annealing. Morgan & Claypool Publishers Series. Morgan & Claypool Publishers, San Rafael (2014)
Lucas, A.: Ising formulations of many NP problems. Front. Phys. 2, 5 (2014)
Acknowledgments
Faisal Shah Khan is indebted to Davide La Torre and Joel Lucero-Bryan for helpful discussion on the topic of fixed-point theorems.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Khan, F.S., Humble, T.S. (2019). Nash Embedding and Equilibrium in Pure Quantum States. In: Feld, S., Linnhoff-Popien, C. (eds) Quantum Technology and Optimization Problems. QTOP 2019. Lecture Notes in Computer Science(), vol 11413. Springer, Cham. https://doi.org/10.1007/978-3-030-14082-3_5
Download citation
DOI: https://doi.org/10.1007/978-3-030-14082-3_5
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-14081-6
Online ISBN: 978-3-030-14082-3
eBook Packages: Computer ScienceComputer Science (R0)